The technique of symplectic
manifolds is well known to provide the adequate Hamiltonian formulation of
autonomous mechanics. Its realistic example is a mechanical system whose configuration
space is a manifold

*M*and whose phase space is the cotangent bundle*T*M*of*M*provided with the canonical symplectic form on*T*M*. Any autonomous Hamiltonian system locally is of this type.
However, this geometric
formulation of autonomous mechanics is not extended to mechanics under
time-dependent transformations because the symplectic form fails to be invariant
under these transformations. As a palliative variant, one has developed time-dependent
mechanics on a configuration space

*Om*

*Q=*where**R**xM**is the time axis. Its phase space***R**R**xT*M*is provided with the pull-back presymplectic form. However, this presymplectic form also is broken by time-dependent transformations.
We address non-relativistic
mechanics in a case of arbitrary time-dependent transformations. Its configuration
space is a fibre bundle

*Q->*. Its velocity space is the first order jet manifold of sections of**R***Q->*. A phase space is the vertical cotangent bundle**R***V*Q*of*Q->*.**R**
This formulation of
non-relativistic mechanics is similar to that of classical field theory on
fibre bundles over a base of dimension >1 difference between mechanics and field
theory however lies in the fact that connections on bundles over

**(#)**. A

**are flat, and they fail to be dynamic variables, but describe reference frames.***R***References:**

**WikipediA**:

**Non-autonomous_mechanics**

**Advanced mechanics. Mathematical introduction**,

**arXiv: 0911.0411**

Очевидно, для Вас механика это математика.

ReplyDeleteДа, хотя и не совсем. Это даже не теоретическая, а математическая физика. И теорфизики и матфизики публикуются в разных журналах. А матфизики и математики тоже публикуются в разных журналах.

ReplyDelete